Workshop report 1. Daniels report is on website 2. Don t expect to write it based on listening to one project (we had 6 only 2 was sufficient

Transcription

1 Workshop report 1. Daniels report is on website 2. Don t expect to write it based on listening to one project (we had 6 only 2 was sufficient quality) 3. I suggest writing it on one presentation. 4. Include figures (from a related paper or their presentation) 5. Include references

2 May 8, CODY Machine Learning for finding oil, focusing on 1) robust seismic denoising/interpolation using structured matrix approximation 2) seismic image clustering and classification, using t-sne(tdistributed stochastic neighbor embedding) and CNN. Weichang Li, Goup Leader Aramco, Houston. May 10, Class HW First distribution of final projects. Ocean acoustic source tracking. Final projects. Final project is the main goal in last month. Bishop Ch 9 Mixture models May 15, CODY Seismology and Machine Learning, Daniel Trugman (half class), ch 8 Graphical models May 17, Class HW ch 8 May 22, Dictionary learning, Mike Bianco (half class), Bishop Ch 13 May 24, Class HW Bishop Ch 13 MAY 30 CODY May 31, No Class. Workshop, Big Data and The Earth Sciences: Grand Challenges Workshop June 5, Discuss workshop, ch13. Spiess Hall open for project discussion 11am-. June 7, Workshop report. No class June 12 Spiess Hall open for project discussion 9-11:30am and 2-7pm June 16 Final report delivered. Beer time For final project discussion every afternoon Mark and I will be available Chapter 13 Sequential data

3 Ocean source tracking X Re-implement Source Localization in an Ocean Waveguide using Supervised Machine Learning X-ray spectrum absorption interpretation using NN Neural decoding Plankton Transfer learning and deep feature extraction for planktonic image data sets Speaker tagger Coral Resturant Amazon rainforest (Kaggle) Myshake Seismic High-precision indoor positioning framework for most wifi-enabled devices Please ask questions Mark and I available all afternoons. Just come or for time slots. Spiess hall 330 is open Monday 5 and 12 June. If interested I can book it at other times Report Rather concise than long. Larger group can do more. Start with some very simple example. To show your idea and that it is working. End with showing the advanced abilities Several figures. Equations are nice. Delivery Zip file (Friday 16) Main code (not all). It should be able to run. Report (pdf preferred). Final Report

4 PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS

5 Graph terminology Before we continue, we must define a few basic terms, most of which are very intuitive. A graph G =(V, E) consists of a set of nodes or vertices, V = {1,...,V}, andaset of edges, E = {(s, t) :s, t V}. We can represent the graph by its adjacency matrix, in which we write G(s, t) =1to denote (s, t) E, thatis,ifs t is an edge in the graph. If G(s, t) =1iff G(t, s) =1,wesaythegraphisundirected, otherwiseitisdirected. We usually assume G(s, s) =0,whichmeanstherearenoself loops. Here are some other terms we will commonly use: Parent For a directed graph, the parents of a node is the set of all nodes that feed into it: pa(s) {t : G(t, s) =1}. Child For a directed graph, the children of a node is the set of all nodes that feed out of it: ch(s) {t : G(s, t) =1}. Family For a directed graph, the family of a node is the node and its parents, fam(s) = {s} pa(s). Root For a directed graph, a root is a node with no parents. Leaf For a directed graph, a leaf is a node with no children. Ancestors For a directed graph, the ancestors are the parents, grand-parents, etc of a node. That is, the ancestors of t is the set of nodes that connect to t via a trail: anc(t) {s : s t}. Descendants For a directed graph, the descendants are the children, grand-children, etc of anode.thatis,thedescendantsofs is the set of nodes that can be reached via trails from s: desc(s) {t : s t}. Neighbors For any graph, we define the neighbors of a node as the set of all immediately connected nodes, nbr(s) {t : G(s, t) =1 G(t, s) =1}. For an undirected graph, we

6 write s t to indicate that s and t are neighbors (so (s, t) E is an edge in the graph). Degree The degree of a node is the number of neighbors. For directed graphs, we speak of the in-degree and out-degree, whichcountthenumberofparentsandchildren. Cycle or loop For any graph, we define a cycle or loop to be a series of nodes such that we can get back to where we started by following edges, s 1 s 2 s n s 1, n 2. Ifthe graph is directed, we may speak of a directed cycle. For example, in Figure 10.1(a), there are no directed cycles, but is an undirected cycle. DAG A directed acyclic graph or DAG is a directed graph with no directed cycles. See Figure 10.1(a) for an example. Topological ordering For a DAG, a topological ordering or total ordering is a numbering of the nodes such that parents have lower numbers than their children. For example, in Figure 10.1(a), we can use (1, 2, 3, 4, 5), or(1, 3, 2, 5, 4), etc. Path or trail A path or trail s t is a series of directed edges leading from s to t. Tree An undirected tree is an undirectecd graph with no cycles. A directed tree is a DAG in which there are no directed cycles. If we allow a node to have multiple parents, we call it a polytree, otherwisewecallitamoraldirectedtree. Forest A forest is a set of trees. Subgraph A(node-induced)subgraph G A is the graph created by using the nodes in A and their corresponding edges, G A =(V A, E A ). Clique For an undirected graph, a clique is a set of nodes that are all neighbors of each other. A maximal clique is a clique which cannot be made any larger without losing the clique property. For example, in Figure 10.1(b), {1, 2} is a clique but it is not maximal, since we can add 3 and still maintain the clique property. In fact, the maximal cliques are as follows: {1, 2, 3}, {2, 3, 4}, {3, 5}.

7 Three types of graphical model Directed graphs useful for designing models Undirected graphs good for some domains, e.g. computer vision Factor graphs useful for inference and learning

8 Bayesian Networks (Bayes Nets) or Directed graphical model (DGM) Decomposition

9 Directed Graphs or Bayesian Networks General Factorization

10 Bayesian Curve Fitting (1) Polynomial Plate

11 Bayesian Curve Fitting (3) Input variables and explicit hyperparameters Condition on data

12 Bayesian Curve Fitting Prediction Predictive distribution: where

13 Generative Models Causal process for generating images

14 Discrete Variables (1) General joint distribution: K 2-1 parameters Independent joint distribution: 2(K-1) parameters General joint distribution over M variables: K M -1parameters M-node Markov chain: K-1 + (M-1) K(K-1) parameters

15 Discrete Variables: Bayesian Parameters (1)

16 Discrete Variables: Bayesian Parameters (2) Shared prior

17 Parameterized Conditional Distributions If K-state variables, has O(K M ) parameters. are discrete, in general The parameterized form requires only M + 1 parameters

18 Conditional Independence a is independent of b given c Equivalently Notation

19 Conditional Independence: Example 1

20 Conditional Independence: Example 2

21 Conditional Independence: Example 3 Note: this is the opposite of Example 1, with c unobserved.

22 D-separation A, B, and C are non-intersecting subsets of nodes in a directed graph. path from A to B is blocked if it contains a node such that either a) the arrows on the path meet either head-to-tail or tail-to-tail at the node, and the node is in the set C, or b) the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C. If all paths from A to B are blocked, A is said to be d-separated from B by C. Then the joint distribution over all variables satisfies. D-separation: Example

23 Markov Random Fields or Undirected Graphs

24 Cliques and Maximal Cliques Clique Maximal Clique

25 Where Joint Distribution is the potential over clique C and is the normalization coefficient; note: M K-state variables K M terms in Z. Energies and the Boltzmann distribution

26 Illustration: Image De-Noising Noisy Image Restored Image (ICM) Restored Image (Graph cuts)

27 Converting Directed to Undirected Graphs (1)

28 Converting Directed to Undirected Graphs (2) Additional links

29 Directed vs. Undirected Graphs (2)

30 Inference in Graphical Models

31 Inference on a Chain

32 Inference on a Chain

33 Inference on a Chain

34 Inference on a Chain To compute local marginals: Compute and store all forward messages,. Compute and store all backward messages,. Compute Z at any node x m Compute for all variables required.

35 Trees Undirected Tree Directed Tree Polytree

36 Factorization Directed graphs: Undirected graphs: Both have the form of products of factors:

37 Factor Graphs More verbose!

38 from Directed Graphs to Factor Graphs

39 Factor Graphs from Undirected Graphs

40 INFERENCE

41 The Sum-Product Algorithm (1) Objective: i. to obtain an efficient, exact inference algorithm for finding marginals; ii. in situations where several marginals are required, to allow computations to be shared efficiently. Key idea: Distributive Law 7 versus 3 operations

42 The Sum-Product Algorithm f 3 (x, y) y u w x f 1 (u, w) f 2 (w, x) f 4 (x, z) z

43 What if the messages are intractable? True distribution Monte Carlo Variational Message Passing Expectation propagation

44 Learning is just inference!

45 The Sum-Product Algorithm (2)

46 The Sum-Product Algorithm (3)

47 The Sum-Product Algorithm (4)

48 The Sum-Product Algorithm (5) Initialization To compute local marginals: Pick an arbitrary node as root Compute and propagate messages from the leaf nodes to the root, storing received messages at every node. Compute and propagate messages from the root to the leaf nodes, storing received messages at every node. Compute the product of received messages at each node for which the marginal is required, and normalize if necessary.

49 Sum-Product: Example (1)

50 Sum-Product: Example (2) and (3)